How to add some blur inside a glass sphere and keep the outside sharp & reflective?
"Blur" is the right word for describing the effect, but the word for the cause of it is "roughness". Roughness simulates a surface that has microscopic bumps on it. Because of these bumps, light refracts with a little bit of randomness, causing a blur.

All you have to do is increase the roughness to about 0.1, in your case:



Roughness will also blur out the reflections. You said that you want the reflections to be sharp. But look carefully at the reference image. The reflections are slightly blurry, too!

Anyway, here is my result of a sphere with the principled shader and a bit of roughness:



Note that if you would actually need a sharp reflection, you would have to mix a rough refraction with a smooth glossy shader, controlling the proportion by the Fresnel node. (Or you could use the clearcoat feature in the principled shader). This would not be realistic, though, because usually the roughness for the reflection and refraction is the same – unless there are multiple layers in the material, which does not seem to be the case for a simple glass sphere.